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We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimisation into a free boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, $${\sigma }$$ , and a measure of the bonding strength between the film and substrate, $${\gamma }$$ . We prove upper bounds on the minimum energy of the form $${\sigma }^a {\gamma }^b$$ and find that there are four different parameter regimes corresponding to different values of a and b and to different folding patterns of the film. In some cases, the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds. [ABSTRACT FROM AUTHOR]

When one slightly pushes a thin elastic sheet at its center into a hollow cylinder, the sheet forms (to a high degree of approximation) a developable cone, or d-cone for short. Here we investigate one particular aspect of d-cones, namely the scaling of the elastic energy with the sheet thickness $$h$$. Following recent work of Brandman et al. () we study the Dirichlet problem of finding the configuration of minimal elastic energy when the boundary values are given by an exact d-cone. We improve their result for the energy scaling. In particular, we show that the deviation from the logarithmic energy scaling is bounded by a constant times the double logarithm of the thickness. [ABSTRACT FROM AUTHOR]

It has been found in numerical experiments [T. Liang and T. A. Witten, Phys. Rev. E (3), 73 (2006), 046604] that when one removes a sector from an elastic sheet and glues the edges of the sector back together, the resulting configuration is radially symmetric and nearly conical. We make a rigorous analysis of this setting under two simplyfying assumptions: First, we only consider radially symmetric configurations. Second, we consider the so-called von Kármán limit, where the size of the removed region as well as the deformations are small. We choose free boundary conditions for a sheet of infinite size. We show existence of minimizers of the suitably renormalized free energy functional. As a by-product, we obtain a lower bound for the elastic energy that has been conjectured in the related context of d-cones [S. Müller and H. Olbermann, Calc. Var. Partial Differential Equations, (2013), 10.1007/s00526-013-0616-6]. Moreover, we determine the shape of minimizers at infinity up to terms that decay like exp(-C√/r). [ABSTRACT FROM AUTHOR]

Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of SO(3). First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of Γ-convergence. [ABSTRACT FROM AUTHOR]

We consider a family of non-convex integral functionals where W is a Carathéodory function periodic in its first variable, and non-degenerate in its second. We prove under suitable conditions that the Γ-limits corresponding to linearization ( h → 0) and homogenization ( $${\varepsilon\rightarrow 0}$$) commute, provided W is minimal at the identity and admits a quadratic Taylor expansion at the identity. Moreover, we show that the homogenized integrand, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the second variation of W. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of h, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von Kármán plate equation. [ABSTRACT FROM AUTHOR]

Since their formulation almost 100 years ago, the von Kármán (vK) plate equations have been frequently used both by engineers and by analysts to study thin elastic bodies, in particular their stability behaviour under applied loads. At the same time the derivation of these equations met some harsh criticism and their precise mathematical status has been unclear until very recently. Following up on a recent variational derivation of the vK theory by Friesecke, James and Müller from three-dimensional nonlinear elasticity we study the predictions and the validity of the vK equation in the presence of in-plane compressive forces. The first main result is a stability alternative: either the load leads to instability already in the nonlinear bending theory of plates (Kirchhoff–Love theory), or it leads to an instability in a geometrically linear KL theory (‘linearized instability’), or vK theory is valid. The second main result states that under suitable conditions the critical loads for nonlinear stability and linearized instability coincide. The third main result asserts this critical load also agrees with the load beyond which the infimum of the vK functional is −∞. The main ingredients are a sharp rigidity estimate for maps with low elastic energy and a study of the properties of isometric immersions from a set in $${\mathbb{R}^2}$$ to $${\mathbb{R}^3}$$ and their geometrically linear counterparts. [ABSTRACT FROM AUTHOR]

We provide existence theorems in nonlinear elasticity for minimum problems modeling the deformations of a crystal with a given dislocation. A key technical difficulty is that due to the presence of a the dislocation the elastic deformation gradient cannot be in L 2. Thus one needs to consider elastic energies with slow growth, to which the original results of Ball cannot be applied directly. [ABSTRACT FROM AUTHOR]

A rigorous derivation is given of a rod theory for a multiphase material, starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Γ-convergence. [ABSTRACT FROM AUTHOR]

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼ h β , where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼ h α . Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α= β=0, nonlinear bending theory to α= β=2, von Kármán theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)3→ℝ3, the L 2 distance of ∇ v from a single rotation is bounded by a multiple of the L 2 distance from the set SO(3) of all rotations. [ABSTRACT FROM AUTHOR]

We study incremental problems in geometrically nonlinear elastoplasticity. Using the multiplicative decomposition Dϕ= FelFpl we consider general energy functionals of the form ℐ(ϕ, Fpl) = ∫ΩU(x, DϕF-1pl, Fpl, &Gscr;(Fpl)) dx-<ℓ, ϕ>, which occur as the sum of the stored energy and the dissipation in one time step. Here &Gscr;(Fpl) is the dislocation tensor which takes the form ${1\over {\rm det} F_{\rm pl}}$ curl3(Fpl),FplT in dimension d = 3. Imposing the usual constraint det Fpl ≡ 1 and suitable growth and polyconvexity conditions on U we show that the minimum of ℐ is attained in the natural Sobolev spaces. Moreover, we are able to treat multiple time steps by controlling the stored and dissipated energies. We also address the relation of the incremental problem to the time-continuous energetic formulation of elastoplasticity. [ABSTRACT FROM AUTHOR]

Using a variational approach we rigorously deduce a nonlinear model for inextensible rods from three-dimensional nonlinear elasticity, passing to the limit as the diameter of the rod goes to zero. The theory obtained is analogous to the Föppl–von Kármán theory for plates. We also derive an asymptotic expansion of the solution and compare it to a similar expansion which Murat and Sili obtained starting from three-dimensional linear elasticity. [Copyright &y& Elsevier]

Abstract: Thin elastic sheets under isotropic compression, such as for example blisters formed by thin films which debonded from the substrate, can exhibit remarkably complex folding patterns. We discuss the scaling of the elastic energy with respect to the film thickness, and show that in certain regimes the optimal energy scaling can be reached by self-similar folding patterns that refine towards the boundary, in agreement with experimental observations. We then extend the analysis to anisotropic compression, and discuss a simplified scalar model which suggests the presence of a transition between a regime where the deformation is governed by global properties of the domain and another one where the direction of maximal compression dominates and the scale of the folds is mainly determined by the distance to the boundary in the direction of the folds themselves. [Copyright &y& Elsevier]

We show that the nonlinear bending theory of shells arises as a Γ-limit of three-dimensional nonlinear elasticity. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]